I am trying to address a specific, but also rather abstract, problem, which briefly can be stated as follows:
Let $f$, $g$ be real-valued functions defined for all column vectors in $\Re^n$, i.e.,
$$ f,g:\Re^n \rightarrow \Re. $$
It would be extremely desired to find a (family of) function(s) $g$ that is bounded as follows:
$$ 0 \leq g(\mathbf{x}) \leq f(\mathbf{x}), \forall \mathbf{x}\in\Re^n. $$
I am particularly interested in those cases for which $f$ is in the following form:
$$ f(\mathbf{x}) = K exp \left\{ -\frac{1}{2}\mathbf{x}^TA\mathbf{x} \right\} $$
I am not sure whether there's a feasible approach to do so or not, while I am not deeply familiarized with issues concerning function bounding.
Could anyone help? Thanks a lot!
Obviously, if we select the $g$ functions to be of the following form
$$ g(\mathbf{x}) = \lambda exp \left\{ -\frac{1}{2} \mathbf{x}^TA\mathbf{x}\right\}, $$
then
$$ 0 \leq g(\mathbf{x}) \leq Kexp \left\{ -\frac{1}{2} \mathbf{x}^TA\mathbf{x}\right\} \Rightarrow 0 \leq \lambda \leq K, $$
which I think is correct, but I would like to find something more sophisticated...
Anyone? Thanks!